Problem: Multiply the following complex numbers, marked as blue dots on the graph: $[2(\cos(\frac{19}{12}\pi) + i \sin(\frac{19}{12}\pi))] \cdot [4(\cos(\frac{1}{3}\pi) + i \sin(\frac{1}{3}\pi))]$ (Your current answer will be plotted in orange.)
Answer: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $2(\cos(\frac{19}{12}\pi) + i \sin(\frac{19}{12}\pi))$ ) has angle $\frac{19}{12}\pi$ and radius $2$ The second number ( $4(\cos(\frac{1}{3}\pi) + i \sin(\frac{1}{3}\pi))$ ) has angle $\frac{1}{3}\pi$ and radius $4$ The radius of the result will be $2 \cdot 4$ , which is $8$ The angle of the result is $\frac{19}{12}\pi + \frac{1}{3}\pi = \frac{23}{12}\pi$ The radius of the result is $8$ and the angle of the result is $\frac{23}{12}\pi$.